H theorem for contact forces in granular materials

TL;DR

This paper develops a maximum entropy theorem for granular contact forces, providing a new theoretical framework that explains force statistics in idealized granular packings without relying on Edwards' hypothesis.

Contribution

It introduces a generalized stosszahlansatz as an alternative to Edwards' hypothesis, deriving contact force statistics from physical assumptions and validating them with simulations.

Findings

The theorem accurately predicts contact force distributions in isotropic packings.

Simulation data shows good agreement with the maximum entropy predictions.

The approach offers a reductionist explanation for granular force statistics.

Abstract

A maximum entropy theorem is developed and tested for granular contact forces. Although it is idealized, describing two dimensional packings of round, rigid, frictionless, cohesionless disks with coordination number Z=4, it appears to describe a central part of the physics present in the more general cases. The theorem does not make the strong claims of Edwards' hypothesis, nor does it rely upon Edwards' hypothesis at any point. Instead, it begins solely from the physical assumption that closed loops of grains are unable to impose strong force correlations around the loop. This statement is shown to be a generalization of Boltzmann's Assumption of Molecular Chaos (his \textit{stosszahlansatz}), allowing for the extra symmetries of granular stress propagation compared to the more limited symmetries of momentum propagation in a thermodynamic system. The theorem that follows from this is similar to Boltzmann's $H$ theorem and is presented as an alternative to Edwards' hypothesis for explaining some granular phenomena. It identifies a very interesting feature of granular packings: if the generalized \textit{stosszahlansatz} is correct, then the bulk of homogeneous granular packings must satisfy a maximum entropy condition simply by virtue of being stable, without any exploration of phase space required. This leads to an independent derivation of the contact force statistics, and these predictions have been compared to numerical simulation data in the isotropic case. The good agreement implies that the generalized \textit{stosszahlansatz} is indeed accurate at least for the isotropic state of the idealized case studied here, and that it is the reductionist explanation for contact force statistics in this case.